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Localised Collective Excitations
in Doped Graphene in a Strong
Magnetic Field
[arXiv:0902.4176]
Andrea M. Fischer1, Alexander B. Dzyubenko2,3 and Rudolf A. Roemer1
1University
of Warwick, Coventry, UK
State University Bakersfield
3General Physics Institute, Russian Academy of Sciences, Moscow
2California
What is Graphene?
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• “Graphene is the name given to a single layer of carbon
atoms densely packed into a benzene-ring structure”
• First isolated in 2004 by physicists at the University of
Manchester, UK and the Institute for Microelectronics
Technology, Chernogolovka, Russia
K.S. Novoselov et al.,
Science 306, 666
(2004)
Andre Geim
Dispersion Relation
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A. H. Castro Neto et al., Rev. Mod. Phys 81, 109 (2009)
vF~106 ms-1
linear dispersion relation
at low energies
P.R. Wallace, Phys. Rev., 71, 622 (1947)
What’s all the fuss?
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• Fundamentally interesting physics and useful physical
properties
• QED in a mesoscopic system – e.g. room temperature
anomalous integer quantum Hall effect
K.S. Novoselov et al, Nature 438, 197
• Important applications i.e. nanoelectronics – is graphene the
new silicon?
Credit: Andre Geim
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Unanswered Questions
• How does substrate affect
graphene properties?
• What is the nature of
disorder in graphene?
• How could an energy gap be
opened at the K points?
- constriction i.e. nanoribbons
- application of an electric
field
Credit: Andre Geim
Aim of Project
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• Single graphene sheet in a strong perpendicular magnetic
field and single axially symmetric charged impurity
- corresponds to a low impurity density
B
High magnetic field
approximation –
reduce disorder
broadening, so LLs
well defined
• Want to find collective excitations localised on impurity
Single Particle Picture
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• Single impurity at origin – symmetric gauge: A=1/2 B x r
• No intervalley scattering
• Corresponding eigensystem:
Quantum numbers: n = …,-1,0,1,…, m=0,1,…, σ = , , s = ↑,↓
• C.f. 2DEG
Self Energy Corrections to Cyclotron Resonance
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• In 2DEG Kohn’s theorem
applies -CR independent of
e-e interactions
• Doesn’t hold for graphene –
e-e interactions very
important
• LL energy renormalised by
self energy corrections due to
exchange with electrons
below Fermi level
• Corrections to CR diverge
Seen experimentally: Z. Jiang
logarithmically with the cut off et al., PRL 98, 197403 (2007)
Collective Excitations
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c † creates electron
Renormalised single
particle energies
Dynamical exchange interaction
Impurity interaction
d † creates hole
Direct electron-hole attraction
π e2
E0 =
2 εℓB
Connection to 2DEG
Energy units
m
a
Ex
es
l
p
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Which transitions are allowed?
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In the dipole approximation, contribution to
Hamiltonian from incoming circularly polarised light is
σ±= σx ± iσy corresponds to
left and right circularly
polarised light
Ϭ+ Ϭ|nfinal|-|ninitial|=±1
mfinal=minitial
Ϭfinal= Ϭinitial
sfinal=sinitial
Which transitions should we take into account?
• Do calculations for sublevel filling factors of the zeroth LL.
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• Infinitely many transitions with same Mz are mixed by Coulomb
interactions
• Only consider mixing for those with the same energies – strong
magnetic field approx
Ϭ+
Ϭ-
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Numerical Considerations
• Matrix comes in 4x4 blocks
• Need to truncate at finite m – ok, because
only seek localised states
• Size limit imposed by calculating matrix
elements with high order Laguerre
polynomials, not by diagonalisation
• Use 4x50=200 basis elements
• Results stable wrt changing matrix size
In the Absence of an Impurity
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A. Iyengar et al. PRB 75, 125430 (2007); Yu. A. Bychkov and G. Martinez, PRB
77, 125417 (2008)
• Use Landau gauge
A = −Byxˆ
• m→ky
• Thickness of band
is 0.75E0 with
π e2
E0 =
2 εℓB
Results
Symmetry relation: Results same under
Mz → -Mz, ν → ν-4, D+ → ΑPDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com
Evolution with Filling Factor
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Summary/Outlook
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• Used a secondary quantised approach to determine collective excitations
bound on an impurity in a single sheet of graphene in the presence of a
strong perpendicular magnetic field [arXiv:0902.4176]
• Could our results be experimentally detected?
Only know relative strengths
Could think about:
- rippling in graphene layer
- intervalley scattering
- nature of impurity
• First step towards combining work on magnetoplasma in pristine graphene
and disordered graphene
• Could they enhance understanding of the nature of disorder in graphene?
• Further work:
Non integer filling factors: consider opposite limit of low electron density,
so D- states are formed.
Thank you for listening
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Rudolf Roemer
University of Warwick, Coventry
Other work:
• Anderson transition in BCC and
FCC lattices using transfer-matrix
method [PRB 77, 245117-8 (2008)]
Alexander Dzyubenko
(California State University,
Bakersfield and General Physics
Institute, Russian Academy of
Sciences, Moscow)
• Excitonic storage in an AharonovBohm nanoring with applied inplane electric field [PRL 102, 096405
(2009)]